Electron correlation radial

The Slater—Condon integrals Ft(ff), Ft(fd), and Gj-(fd), which represent the static electron correlation within the 4f" and 4f 15d1 configurations. They are obtained from the radial wave functions R, of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23,31... 

Electron correlation has been taken care of by using multiple exponent Hylleraas type basis sets. The effect of radial and full correlation on such affinity is analyzed and is reflected in Figure 30. 

SL-BOVB Calculations of F2 The L-BOVB wave function can be further improved by incorporating radial electron correlation for the active electrons... 

As mentioned previously, parameter a may be viewed as the effective nuclear charge felt by either one of the two electrons. Such an interaction is commonly called the screening effect. Furthermore, as described by this wavefunction, the two electrons move independently of each other, i.e., angular correlation is ignored. Electron correlation may be taken as the tendency of the electrons to avoid each other. For helium, angular correlation describes the two electrons inclination to be on opposite sides of the nucleus. On the other hand, radial correlation, or screening effect, is the tendency for one electron to be closer to the nucleus, while the other one is farther away. A one-parameter trial function that does take angular correlation into account is... 

In compounds containing heavy main group elements, electron correlation depends on the particular spin-orbit component. The jj coupled 6p j2 and 6/73/2 orbitals of thallium, for example, exhibit very different radial amplitudes (Figure 13). As a consequence, electron correlation in the p shell, which has been computed at the spin-free level, is not transferable to the spin-orbit coupled case. This feature is named spin-polarization. It is best recovered in spin-orbit Cl procedures where electron correlation and spin-orbit interaction can be treated on the same footing—in principle at least. As illustrated below, complications arise when configuration selection is necessary to reduce the size of the Cl space. The relativistic contraction of the thallium 6s orbital, on the other hand, is mainly covered by scalar relativistic effects. 

Fig. 5.63. Electron-Cs" and electron-f radial pair correlation functions. (Reprinted from C. Malesdo, Mol. Phys. 69 895,1990.)...

Several approaches were pursued in the process of finding an interpretation more physically intuitive than the somewhat hollow, ex post facto interpretation of Eq. (1), that is, that the independent-particle picture represented by a single configuration is spoiled by electron-electron correlation, particularly by angular correlation, because the second configuration leaves the radial distribution relatively unaffected but changes the angular distribution. 

Within the hyperspherical method, new quantum numbers K, T and A are introduced to describe two-electron correlations. Both K and T are angular correlation numbers (omitted here for simplicity, see [333]), while A = 0, 1 is a radial quantum number, often written as 0,+,— because it is related to the + and — classification of Cooper, Fano and Pratts [323] described in section 7.10. Another quantum number which is often used is v = n — 1 — K — T, where n is the principal quantum number. The number v turns out to be the vibrational quantum number of the three-body system, or the number of nodes contained between the position vectors ri and r2 of the two electrons [334]. 

Because of the complexity of the PHF function, only very small electronic systems were initially considered. As first example, the electronic energy of some four electron atomic systems was calculated using the Brillouin procedure [8]. For this purpose, a short double zeta STO basis set. Is, Is , 2s and 2s , with optimized exponents was used. The energy values obtained are given in Table 1. In the same table, the RHF energy values calculated with the same basis are gathered for comparison. It is seen that the PHF model introduces some electronic correlation in the wave-function. Because of the nature of the basis set formed by only s-type orbitals, only radial correlation is included which account for about 30% of the electronic correlation energy. 

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